To Iterate or Not to Iterate?

As far as I can tell, this problem has to be solved using an iterative approach. Newton's method is a great way to do it, and there is a good Brilliant page on the subject.

Newton's Method

Here's what the result looks like in Excel:

To me, it's a much more interesting exercise to find the complex solutions. There is a multivariate version of Newton's Method that I have used to find complex solutions. Instead of a single derivative, the method employs a matrix of partial derivatives, known as a Jacobian matrix.

Here's a plot of the $A+Bi$ solutions in the complex plane. Note that the single real-valued solution is included. This plot was obtained by running the iterative multivariate algorithm numerous times, with different random starting values for $A$ and $B$ each time. The particular random starting pair determines which solution the algorithm converges to. The Python code is attached for reference.

This is only for the real solutions.

First, do the substitution $x = - u$

Doing this makes our original equation from $x + e^{x} = 0$ to $-u + e^{-u} = 0$

Rearranging the terms gives, $e^{-u} = u$

Taking a natural log on both sides gives, $ln(u) = - u$

Plotting the graphs gives the solution $u = 0.567$

And hence $\boxed{x = -0.567}$